we will be able to gather the infinitely many copies of \(\mathcal {H}_k\)  into one irreducible representation for the Lie algebra \(\mathfrak {sl}_h(2)\) . Consequentially, the decomposition becomes multiplicity-free under the joint action of the Howe dual pair \((\mathsf {SO}(m),\mathfrak {sl}(2))\) , see [1]} for more details on dual pairs. The `gathering' mentioned above is done by introducing infinite-dimensional Verma modules for the Lie algebra \(\mathfrak {sl}(2)\) . As these will play a key role in this paper, we briefly recall their construction.
